Bring 4 quarts of water and 1/2 teaspoon salt to a boil in a large pot. Add the spaghetti and cook according to package directions until al dente, about 5–7 minutes; drain well in a colander.
In a large bowl, whisk together the chicken broth, creamy peanut butter, honey, soy sauce, red chili flakes, ground ginger, and minced garlic until smooth and well combined.
Add the drained noodles, cooked chicken, and sliced carrots to the bowl with the sauce. Toss thoroughly until everything is evenly coated with the peanut sauce.
Transfer to a serving bowl and sprinkle with thinly sliced scallions and chopped peanuts before serving.